In this paper, we study {it operator spaces/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $Esubset B(H)$, with $H$ Hilbert. We will be mainly concerned here with the ``geometry'' of {it finite dimensional/} operator spaces. In the Banach space category, it is well known that every separable space embeds isometrically into $ell_infty$. Moreover, if $E$ is a finite dimensional normed space then for each $vp>0$, there is an integer $n$ and a subspace $Fsubset ell^n_infty$ which is $(1+vp)$-isomorphic to $E$, i.e. there is an isomorphism $ucolon E o F$ such that $|u| |u^{-1}|le 1+vp$. Here of course, $n$ depends on $vp$, say $n=n(vp)$ and usually (for instance if $E = ell^k_2$) we have $n(vp) o infty$ when $vp o 0$. Quite interestingly, it turns out that this fact is not valid in the category of operator spaces: although every operator space embeds completely isometrically into $B(H)$ (the non-commutative analogue of $ell_infty$) it is not true that a finite dimensional operator space must be close to a subspace of $M_n$ (the non-commutative analogue of $ell^n_infty$) for some $n$. The main object of this paper is to study this phenomenon.
